NAG FL Interface
c09ccf (dim1_​multi_​fwd)

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1 Purpose

c09ccf computes the one-dimensional multi-level discrete wavelet transform (DWT). The initialization routine c09aaf must be called first to set up the DWT options.

2 Specification

Fortran Interface
Subroutine c09ccf ( n, x, lenc, c, nwl, dwtlev, icomm, ifail)
Integer, Intent (In) :: n, lenc, nwl
Integer, Intent (Inout) :: icomm(100), ifail
Integer, Intent (Out) :: dwtlev(nwl+1)
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Out) :: c(lenc)
C Header Interface
#include <nag.h>
void  c09ccf_ (const Integer *n, const double x[], const Integer *lenc, double c[], const Integer *nwl, Integer dwtlev[], Integer icomm[], Integer *ifail)
The routine may be called by the names c09ccf or nagf_wav_dim1_multi_fwd.

3 Description

c09ccf computes the multi-level DWT of one-dimensional data. For a given wavelet and end extension method, c09ccf will compute a multi-level transform of a data array, xi, for i=1,2,,n, using a specified number, nfwd, of levels. The number of levels specified, nfwd, must be no more than the value lmax returned in nwlmax by the initialization routine c09aaf for the given problem. The transform is returned as a set of coefficients for the different levels (packed into a single array) and a representation of the multi-level structure.
The notation used here assigns level 0 to the input dataset, x, with level 1 being the first set of coefficients computed, with the detail coefficients, d1, being stored while the approximation coefficients, a1, are used as the input to a repeat of the wavelet transform. This process is continued until, at level nfwd, both the detail coefficients, dnfwd, and the approximation coefficients, anfwd are retained. The output array, C, stores these sets of coefficients in reverse order, starting with anfwd followed by dnfwd,dnfwd-1,,d1.

4 References


5 Arguments

1: n Integer Input
On entry: the number of elements, n, in the data array x.
Constraint: this must be the same as the value n passed to the initialization routine c09aaf.
2: x(n) Real (Kind=nag_wp) array Input
On entry: x contains the one-dimensional input dataset xi, for i=1,2,,n.
3: lenc Integer Input
On entry: the dimension of the array c as declared in the (sub)program from which c09ccf is called. c must be large enough to contain the number, nc, of wavelet coefficients. The maximum value of nc is returned in nwc by the call to the initialization routine c09aaf and corresponds to the DWT being continued for the maximum number of levels possible for the given data set. When the number of levels, nfwd, is chosen to be less than the maximum, then nc is correspondingly smaller and lenc can be reduced by noting that the number of coefficients at each level is given by n¯/2 for mode='P' in c09aaf and (n¯+nf-1)/2 for mode='H', 'W' or 'Z', where n¯ is the number of input data at that level and nf is the filter length provided by the call to c09aaf. At the final level the storage is doubled to contain the set of approximation coefficients.
Constraint: lencnc, where nc is the number of approximation and detail coefficients that correspond to a transform with nwlmax levels.
4: c(lenc) Real (Kind=nag_wp) array Output
On exit: let q(i) denote the number of coefficients (of each type) produced by the wavelet transform at level i, for i=nfwd,nfwd-1,,1. These values are returned in dwtlev. Setting k1=q(nfwd) and kj+1=kj+q(nfwd-j+1), for j=1,2,,nfwd, the coefficients are stored as follows:
c(i), for i=1,2,,k1
Contains the level nfwd approximation coefficients, anfwd.
c(i), for i=k1+1,,k2
Contains the level nfwd detail coefficients dnfwd.
c(i), for i=kj+1,,kj+1
Contains the level nfwd-j+1 detail coefficients, for j=2,3,,nfwd.
5: nwl Integer Input
On entry: the number of levels, nfwd, in the multi-level resolution to be performed.
Constraint: 1nwllmax, where lmax is the value returned in nwlmax (the maximum number of levels) by the call to the initialization routine c09aaf.
6: dwtlev(nwl+1) Integer array Output
On exit: the number of transform coefficients at each level. dwtlev(1) and dwtlev(2) contain the number, q(nfwd), of approximation and detail coefficients respectively, for the final level of resolution (these are equal); dwtlev(i) contains the number of detail coefficients, q(nfwd-i+2), for the (nfwd-i+2)th level, for i=3,4,,nfwd+1.
7: icomm(100) Integer array Communication Array
On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization routine c09aaf.
On exit: contains additional information on the computed transform.
8: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, n is inconsistent with the value passed to the initialization routine: n=value, n should be value.
On entry, lenc is set too small: lenc=value.
Constraint: lencvalue.
On entry, nwl=value.
Constraint: nwl1.
On entry, nwl is larger than the maximum number of levels returned by the initialization routine: nwl=value, maximum =value.
Either the initialization routine has not been called first or array icomm has been corrupted.
Either the initialization routine was called with wtrans='S' or array icomm has been corrupted.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The accuracy of the wavelet transform depends only on the floating-point operations used in the convolution and downsampling and should thus be close to machine precision.

8 Parallelism and Performance

c09ccf is not threaded in any implementation.

9 Further Comments

The wavelet coefficients at each level can be extracted from the output array c using the information contained in dwtlev on exit (see the descriptions of c and dwtlev in Section 5). For example, given an input data set, x, denoising can be carried out by applying a thresholding operation to the detail coefficients at every level. The elements c(i), for i = k1+1 ,, k nfwd + 1 , as described in Section 5, contain the detail coefficients, d^ij, for i=nfwd,nfwd-1,,1 and j=1,2,,q(i), where d^ij=dij+σεij and σεij is the transformed noise term. If some threshold parameter α is chosen, a simple hard thresholding rule can be applied as
d¯ ij = { 0, if ​ |d^ij| α d^ij , if ​ |d^ij| > α,  
taking d¯ij to be an approximation to the required detail coefficient without noise, dij. The resulting coefficients can then be used as input to c09cdf in order to reconstruct the denoised signal.
See the references given in the introduction to this chapter for a more complete account of wavelet denoising and other applications.

10 Example

This example performs a multi-level resolution of a dataset using the Daubechies wavelet (see wavnam='DB4' in c09aaf) using zero end extensions, the number of levels of resolution, the number of coefficients in each level and the coefficients themselves are reused. The original dataset is then reconstructed using c09cdf.

10.1 Program Text

Program Text (c09ccfe.f90)

10.2 Program Data

Program Data (c09ccfe.d)

10.3 Program Results

Program Results (c09ccfe.r)